Arizona’s tough immigration law is just the beginning of the conservative battle to clamp down on illegal immigrants. A broader fight is coming—possibly even to change the U.S. Constitution. Sen. Lindsey Graham made headlines last week, telling Fox News he’s considering a constitutional change to revise the right, enshrined in the 14th Amendment, that grants automatic citizenship to any child born in the United States. He would revoke this right for children of illegal immigrants. “It’s called ‘drop and leave,’ ” Graham said, and while he added that he wanted to be “humane,” he also feared that in 20 years’ time, 20 million more such children would be granted citizenship.

Graham wasn’t the first to call for such a measure, but his voice could add credibility to what had been a fringe idea. The South Carolina senator has been a GOP moderate on immigration, willing to work with Democrats to provide a path to citizenship for millions of illegal U.S. residents. But now Graham is speaking the language of organizations like the Federation for American Immigration Reform, which argues against the rights of so-called anchor babies, saying they drain taxpayer dollars. In 2009, then–Georgia Rep. Nathan Deal introduced a bill that would grant citizenship to such infants only if one parent is already a citizen or a legal immigrant. The bill has 92 co-sponsors but has languished in the House.

Some thought Graham was just trying to score short-term political points. Critics point out that, among other things, a new constitutional amendment would need two-thirds support in Congress and ratification by three quarters of the states. But Kevin Bishop, a spokesman for Graham, says that “he is very, very serious about this.” Bishop adds that a new constitutional amendment may not be necessary to achieve the desired outcome, and points to the efforts of Rep. Lamar Smith of Texas. Smith argues that granting automatic citizenship to illegal aliens is a misinterpretation of the 14th Amendment, and that the Constitution gives Congress the power to decide national immigration policies. He believes the issue can be addressed by statute and is enthusiastic about Graham’s interest in getting something done: “He has taken high-profile positions on immigration,” the congressman tells NEWSWEEK, “so his support for an end to birthright citizenship is very significant.”